Download Chapter 10: Confidence Intervals

yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Bootstrapping (statistics) wikipedia, lookup

History of statistics wikipedia, lookup

Confidence interval wikipedia, lookup

Eigenstate thermalization hypothesis wikipedia, lookup

Psychometrics wikipedia, lookup

Foundations of statistics wikipedia, lookup

Statistical hypothesis testing wikipedia, lookup

Resampling (statistics) wikipedia, lookup

Student's t-test wikipedia, lookup

Misuse of statistics wikipedia, lookup

Chapter 11: Significance Testing
Section 1: The Basics
Significance Testing:
◦ A formal procedure for comparing observed data
with a hypothesis whose truth value we want to
◦ A statement about a population parameter, like µ or
Let’s think…
Several cities have begun to monitor and examine
paramedic response times. In Hurtville, the mean
response time to all accidents involving life
threatening accidents was µ=6.7 minutes. The city
manager encourages them to do better each year.
The following year the city manager selects a SRS
of 400 life threatening calls. The average response
time for this sample was 6.48 minutes.
Do these data points provide good evidence that the
response times have decreased since last year?
Let’s think…
What would we want to test?
◦ Normality?
 Standard deviation would be…
◦ For or Against claim?
 Against the claim!
Types of Hypotheses
Null Hypothesis, H0:
◦ The statement we are testing
◦ Statement of “no effect,” “no difference,” or
no change in historical values
Alternative Hypothesis, H1 or Ha:
◦ The claim about the population we are trying
to find evidence for
◦ Suggests that something has changed or
different than expected
Looking at the paramedics…
If we look back at the response times for
our paramedics, what would be our null
hypothesis and alternative hypothesis?
H0 :
◦ Ha :
◦ This is a one-sided alternative since we are
only concerned with deviations in one
Example 1:
Larry’s car averages 26 mpg on the highway. He switches
to a brand of new motor oil that is advertised to
increase gas mileage. After driving 3000 highway miles
with the new oil, he wants to determine if the average
gas mileage has increased.
What are the appropriate null and alternative hypotheses?
Be sure to use the appropriate parameters and define them.
◦ µ = mean gas mileage for Larry’s car on the highway
H0 :
Ha :
Example 2:
A May 2005 Gallup Poll report on a national survey of
1208 teens revealed 72% of teens said they rarely or
never argue with their friends.You wonder whether this
would be true in your school so you conduct your own
survey of random sample of students in your school.
What are the appropriate null and alternative hypotheses?
Be sure to use the appropriate parameters and define them.
◦ p = the proportion of teens in your school who rarely argue with their
H0 :
Ha :
Test Statistics
Compares the value of the parameter as
stated in the null hypothesis with an
estimate of the parameter from the
sample data.
Values of the estimate far from the
parameter value in the direction specified
by the alternate hypothesis give evidence
against the null.
Test Statistics
To assess how far from the parameter, we
standardize the estimate:
Looking at the paramedics…
Given the paramedic example where the
average for the sample of 400 calls was
6.48, find the test statistic we would want
to use to test our hypotheses.
H0 :
Ha :
The probability, assuming H0 is true, that the
observed outcome would take a value as
extreme as or more extreme than actually
The smaller the p-value, the stronger the
evidence is against H0 provided by the data
(ie observed result is unlikely to occur when
H0 is true).
Larger p-values fail to give evidence against
Looking at the paramedics…
H0 :
Ha :
What is the p-value associated with
testing our null hypothesis?
◦ Using our test statistic:
◦ The p-value would be:
So…are we for or against our H0 ?
◦ Small p-value provides strong evidence against H0
Example 3:
Does the job satisfaction of assembly workers differ when their
work is machine-paced rather than self-paced? Assume the
standard deviation of this normal distribution is 60. If there is no
difference in job satisfaction, then the mean is 0. One study chose
18 subjects at random from a group of people who assembled
electronics. These workers gave a mean of 17 (that is these
workers preferred the self-paced environment).
Hypotheses: H0 :
Test Statistic:
Ha :
Statistical Significance
If the p-value is as small as or smaller than
the alpha, we say that the data as
statistically significant at level
At a 95% confidence level,
Note: Significant in a statistical sense, does
not mean “important.” It just means
simply not likely to happen just by chance.
Statistical Significance
If the P = .03 under a 95% confidence
level, then
So is the result statistically significant?
◦ Yes
What about for
◦ No
So…we find the p-value… and we know
if it’s statistically significant…now what…
We want to reject H0 or fail to reject H0
◦ If p-value >
then we fail to reject H0
◦ If p-value <
then we reject H0
Steps for Significance Tests
Hypothesis: Identify the population and
Conditions: Choose the appropriate inference
Calculations: Find test statistic and p-value
Interpret: Decide statistical significance and
parameter you want to draw conclusion about.
State the hypothesis
procedure.Verify conditions.
write conclusion.
Alternative Hypothesis, Ha
Fail to reject H0 :
◦ We cannot find evidence to suggest Ha is
◦ Thus for 2 tail, we cannot find convincing
evidence that the mean differs
◦ Only means the data are consistent with H0,
not that we have evidence that H0 is true.
◦ Want to make statement that “since the
obtained p-value is so large, we don’t have
strong evidence against H0 ”